Math130 Finite Mathematics

Project Two

How to invest the money you manage? -- Linear Programming

Description:

Suppose you are a portfolio manager who overlooks six million dollars. In a simplified situation, you have two choices to invest: high quality bonds and junk bonds. With some restrictions, how should you allocate your money to maximize your return?

In this project, you will learn the concepts of bonds, high quality bonds, and junk bonds; and understand their advantages and disadvantages. Then you will set up a linear programming problem and solve the question proposed in the first paragraph.

You will further study the theory of linear programming and understand how the slope of the objective function changes the optimal solution. Use Desmos to draw the graphs.

Instruction:

1. (1)Understand bonds.

Read: https://www.investor.gov/introduction-investing/investing-basics/investment-products/bonds-or-fixed-income-products/bonds

1. (2)Google search high quality bonds and junk bonds. Give their definitions and several examples.

1. (3)Why are junk bonds called high-yield bonds?

1. (4)As a portfolio manager, you want to maximize your profit. However, you also want to control the risk. The policy of the investment you adopt is to invest at least twice the amount of money in premium-quality bonds as in junk bonds. Google search "at least twice as much" and understand what it means.

1. (5)Suppose you have a total of 6 million dollars to invest. Give all the constraints and draw the feasible region. (Let x be the amount invested in the junk bonds, and y be the amount invested in the premium-quality bonds.)

1. (6)Suppose that the junk bonds have an average yield of 12% and the premium-quality bonds yield 7%. Find the maximum yield.

1. (7)Read the textbook (Section 3.2, page 128-132), understand the theory of linear programming.

1. (8)Using the maximum yield, draw the line of the objective function. Name it L₀.

1. (9)Draw two lines parallel to L₀: one below L₀, (name it L₁) and the other above L₀, (name it L₂). Explain why L₁ and L₂ are not the solutions of the problem. Try to understand the theorem using these lines.

1. (10)Let's assume hypothetically that the junk bonds have an average yield of 7% and the premium-quality bonds yield 12%. Find the maximum yield.

1. (11)Draw another line that represents the new optimal solution. Notice how the new slope changes the solution.

1. (12)Assume hypothetically again that the junk bonds and the premium-quality bonds have the same average yield of 10%. Find the maximum yield.

1. (13)Repeat (11). What did you observe?

Math 130 Name:

Project Two Report Date:

1. 1.(2 points) Briefly talk about what a bond is.

1. 2.(6 points) Give the definitions of high-quality bonds and junk bonds, and several examples.

1. 3.(2 points) Why are junk-bonds called high-yield bonds?

1. 4.(2 points) Let x be the money amount you want to invest in the junk bonds and y the high-quality bonds. Other than x>=0 and y>=0, give another obvious constraint.

1. 5.(4 points) Based on (4) in the instruction, give the second key constraint.

1. 6.(4 points) Draw and shade the feasible region. (If you use desmos.com, try Shift-Screen-s for windows and Shift-Command-4 for mac to take a screen shot.)

1. 7.(8 points) Give the objective function and find the maximum return.

1. 8.(2 points) Copy and paste a theorem that shows the theory of linear programming.

1. 9.(10 points) Copy the graph in 6 then add three lines L₀, L₁ and L₂ on the graph. Explain why L₁ and L₂ are not the solutions of the problem.

1. 10.(4 points) Assume hypothetically that the junk bonds have an average yield of 7% and the premium-quality bonds yield 12%. Find the maximum yield.

1. 11.(6 points) Copy the graph in 6 then add a line representing the optimal solution in 10.

1. 12.(4 points) Assume hypothetically again that the junk bonds and the premium-quality bonds have the same average yield of 10%. Find the maximum yield.

1. 13.(6 points) Repeat 11. Answer the following question: For what values of x and y does the objective function achieve its maximum yield?